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Laboratoire national de métrologie et d’essais

P. Ceriaa, S. Ducourtieuxa, Y. Boukellala, N. Feltina,

A. Allardb and N. Fischer b

a LNE, Nanometrology

b LNE, Mathematics and Statistics Department

Modeling of a metrological AFM

interferometric position

measurement system to determine

its measurement uncertainty

JRP 6DoF : metrology for movement and

positioning in six degrees of freedom

Content

1. Introduction

2. The principal features of the model

3. The used statistical tools

4. The link with the experimental data

5. The most significant results

6. Conclusion and perspectives

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Introduction

Why do we develop a model of the LNE’s metrological AFM ?

The development of the instrument started in 2007. We have now reached the

final phase where we have to deal with the evaluation of the uncertainty budget for

the positioning of the instrument. But the task is not so easy when the specific

metrology loop of the instrument is considered !

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Sebastien Ducourtieux and Benoit Poyet, Meas. Sci. Technol. 22 (2011) 094010

Introduction

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x

y

z

The metrology loop is composed of 4 differential interferometers (Renishaw), 2

zerodur prisms (one linked to the tip, the other one to the sample) and 8 mirrors.

Introduction

Original configuration for the interferometers with very good performances :

- low noise : 0,3 nm at 1,5 MHz

- stability estimated to only few nanometers over one hour.

BUT it is difficult to feel how much sensitive the system is to factors like :

- Parasitic rotations, interferometer misalignment (Abbe error)

- Cosine error

- Non perfect geometry of the prisms (angles of the mirror, shapes and

roughness of the mirror)

- Positioning error

- Basic dilatations

-…

=> This is the reason why we came to a virtual representation of the metrology

system in order to better estimate the uncertainty budget of the instrument and

also to gain knowledge on its behavior.

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Principal features of the model

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How do we model the metrology system ?

The model is developed under Matlab using object oriented programming.

- Mirrors are modeled by a 3D cloud of points

1) plane mirror

2) filters to produce

random or predefined

shape and roughness on

the mirror

3) translation and rotation of the

mirror using homogeneous

coordinates formalism

Principal features of the model

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The prism is simply created by placing each mirror at its correct position

according to mAFM CAD drawing. Each prism is then considered as an object

which can be also translated and rotated using the homogeneous coordinates

formalism.

The interferometers are considered as a black box which only measures the

distance separating the interferometer head and the mirror. Each beam possesses

a source point, a direction vector and a Gaussian lateral extension. The relative

displacement is evaluated according to the following formulas :

Principal features of the model

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distance between the beam axis

and the considered point on the

mirror

Principal features of the model

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A set of about 100 parameters are used to control the model and define the

geometry of the metrology system. Some of these parameters are then coupled

by using other parameters or laws to define some specific behavior (i.e.

homothetic dilatation of the prism or structural frame). Each parameter is

associated with a distribution law and an uncertainty according to our actual

knowledge and our experimental data.

We feed the model with experimental data

Link with experimental data

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Flatness and roughness measurement

using interferometric microscope*

Angular relations between the prism mirror

using the LNE angular platen and

using theodolite system*

*Performed by the Optic Group team from Soleil Synchrotron

Beam intensity profile

Stage Parasitic rotations

Instrument and room thermal stability

Model

Principal features of the model

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GUI

We use Monte Carlo method to propagate the probability density function

associated to each parameter inside the model and then to evaluate the position

measurement uncertainty.

Implemented statistical tools

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10 min for 105 draws and

100 parameters on the

cluster

Quick overview of the results

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Quick overview of the results

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XYZ positioning uncertainty with

the actual set of parameters (not

the definitive one) and for the full

range of displacement

(60 x 60 x 15 µm3):

about 6 nm

Morri’s plan is used to identify the most influential components by evaluating their

interactions two by two :

- Influential due to many interactions with other components (increasing )

- Influential because of the strength of the effect (increasing )

Quick overview of the results

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Mean effect

Standard deviation of the effect

More effect

More interaction

Quick overview of the results

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Mean effect

Standard deviation of the effect

More effect

More interaction

Most influential parameters with the actual set of parameters (not the definitive

one) and for the full range of displacement (60 x 60 x 15 µm3) :

- Parasitic rotations (high mean effect and high interaction)

- Abbe offset (high mean effect and high interaction)

- Sample thickness (high mean effect and high interaction)

- Orthogonally error (high mean effect but no interaction)

Morri’s plan is used to identify the most influential components by evaluating their

interactions two by two :

- Influential due to many interactions with other components (increasing )

- Influential because of the strength of the effect (increasing )

Sobol’ indices are used evaluate the contribution of each parameter to the global

uncertainty :

Quick overview of the results

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Identification of the contribution of each parameter to the global uncertainty using

Sobol’ indices :

Quick overview of the results

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Contribution to the global uncertainty in the actual configuration :

- 75 % coming from Abbe error (parasitic rotations, Abbe Offset and sample

height)

- 24 % coming from the orthogonality error

Conclusion

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- Uncertainty is mainly impacted by Abbe error and orthogonality error

=> for orthogonality, we can introduce corrections

=> for Abbe error, we must improve beam alignment and if possible, bring

some modifications to the stage to reduce parasitic rotations.

- In the actual configuration with the actual set of parameters, the reached

positioning uncertainty for the LNE’s mAFM is 6 nm for the full range of

displacement (60 x 60 x 15 µm3) , but the definitive experimental data are not

used here so it will be reduced !

- Easily reconfigurable model that we can adapted to another configuration or

another instrument (for example to evaluate instruments implementing 6 DoF

displacement and/or 6 DoF metrology loop).

Perspective

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Now, we are working on the modeling of the image construction.

We add a modeled grating and a modeled AFM tip

Objective : to move from the positioning uncertainty to the measurement

uncertainty of standard dimensional properties (i.e. pitch and step height).

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Thank you for your attention

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Thank you for your attention

Implemented statistical tools

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